UK ATC training is (HD... and has been for at least ten years) based on continuous assessment; therefore it is necessary to undertake a recognised training course (mixture of practical and theoretical assessment). These are available at the CAA College at Bournemouth and a couple of other companies (SERCo, Bae Systems at Dundridge etc... I may not be up-to-date as availability changes. These companies course may also open the door to employment ooportunities abroad?)
Whilst you can pay for your own place on course, you will be required to validate your CAA license (achieve an operational endorsement) within a given timescale. Pay your own way and you may struggle to find a post-graduate opportunity to validate.
NATS gets most of its controllers from the Bournemouth output (I believe?) and as you say is the "path most trodden" because few if any, regional airport operators openly advertise for non-qualified personnel... Some have accepted people in the past and sponsored them through training (normally on a shared cost plus return of service agreement)... and may do again, but it is not the norm.
The implimentation of ESARR5 licensing requriements may open the market in the future and allow non-UK controllers (i.e. without a CAA license) to move into UK ATC units and UK personnel to obtain training at foreign colleges when commonality of licenses across the EU is achieved.... (speculative?). There's plenty of information out there on the web.